Quadratic Control Problem Research Articles

On the utopian approach to the multiobjective LQ problem

AbstractThe infinite horizon, multiobjective linear quadratic control problem for continuous time systems is considered. Following a utopian approach, the optimal solution is defined as the solution that minimizes the distance from the utopian point in the cost space. It is shown that under standard stabilizability and detectability assumptions the optimal solution always exists. The solution coincides with the solution of a scalar LQ problem in which the cost matrices are given by a suitably weighted sum of their counterparts in the individual criteria. By exploiting the characterization and properties of the optimal solution, it is shown that the problem can be efficiently solved by means of a Newton‐type algorithm.

Noniterative Approximations to the Solution of the Matrix Riccati Differential Equation

The object of this paper is to present new approximation schemes for the nonlinear matrix Riccati differential equation. They are obtained from any one-step or multistep method applied to the original linear quadratic control problem. They lead to the same type of schemes. However only one matrix inversion is required at each discretization node even if the Riccati equation is a nonlinear equation. This important computational advantage is obtained without altering the original nodal asymptotic convergence rate. It is proved that this rate is the same as the one of the initially chosen scheme.

Distributed control and simulation of a Bernoulli-Euler beam

The optima] distributed control of a Bernoulli-Euler beam is studied. A distributed parameter model of the system dynamics is derived, and this model is used to solve the linear quadratic regulator control problem. The exact model avoids the potentially destabilizing effects of control spillover, as it reproduces the beam dynamics at all frequencies, insofar as the mathematical model represents the actual physical structure. Functional Riccati equations are derived, and numerical procedures are developed to iteratively converge on a solution. The solutions represent gain surfaces, which relate the measured state variables to the control forces at arbitrary points along the beam. An efficient procedure for simulating the response of the closed-loop system to initial conditions and disturbance forces is then developed. Simulation results are presented that indicate that the control law successfully suppresses structural vibration. Issues concerning implementation of the distributed control system are addressed. Finally, an extension of the formulation to multiple beam systems, such as space frames, is discussed.

Two Families of Approximation Schemes for Delay Systems

Based on the Trotter-Kato approximation theorem for strongly continuous semigroups we develop a general framework for the approximation of delay systems. Using this general framework we construct two families of concrete approximation schemes. Approximation of the state is done by functions which are piecewise polynomials on a mesh (m-th order splines of deficiency m). For the two families we also prove convergence of the adjoint semigroups and uniform exponential stability, properties which are essential for approximation of linear quadratic control problems involving delay systems. The characteristic matrix of the delay system is in both cases approximated by matrices of the same structure but with the exponential function replaced by approximations where Pade fractions in the main diagonal resp. in the diagonal below the main diagonal of the Pade table for the exponential function play an essential role.

Optimal control of linear parabolic equation:the constrained right-hand side as control function

The quadratic cost control problem for a linear parabolic equation controlled by the constrained right-hand side of the equation is considered. Finite-dimensional approximation scheme for finding approximate solutions is proposed and its convergence rate is estimated. The set of optimal controls and its continuity properties with respect to the problem's data are studied. The convergence of the approximation scheme in the case when the control consists of both the constrained initial condition and the constrained right-hand side of the equation is proved.

Robust adaptive LQ control schemes

The certainty equivalence principle is used to combine a robust adaptive law with a control structure derived from the linear quadratic (LQ) control problem. The resulting adaptive control scheme is applicable to minimum and nonminimum phase continuous-time plants and is robust with respect to unmodeled dynamics and bounded disturbances. The computational complexity of the continuous-time adaptive LQ control scheme is improved by using a hybrid adaptive law which requires the solution of an algebraic Riccati equation at each interval of time rather than at each time t. >

Unbounded Solutions to the Linear Quadratic Control Problem

Examples are presented to show that the solution of the operational algebraic Riccati equation can be an unbounded operator for infinite dimensional systems in a Hilbert space even with bounded control and observation operators. This phenomenon is connected to the presence of a continuous spectrum in one of the operators. The object of this paper is to fill up the gap in the classical linear quadratic theory. The key step is the introduction of the set of stabilizable initial conditions. Then a new simple approach to the linear-quadratic problem is presented that provides the connection with the notion of approximate stabilizability for the triplet $(A,B,C)$.

Dual linear quadratic optimal control for two-point boundary value descriptor systems

The dual linear quadratic opiimal control problem for the two-point boundary value case of discrete descriptor systems is investigated. It is shown that, in the case where the cost functional docs not have cross terms, the optimal control law can be expressed as an affine function of the stale of the non-causal subsystem, while the optimal cost-to-go can be expressed as a general quadratic expression of the state of the non-causal subsystem.

Study of the discrete singularly perturbed linear-quadratic control problem by a bilinear transformation

This paper presents a new approach in the study of the linear quadratic control problem of singularly perturbed discrete systems. By applying a bilinear transformation, the algebraic discrete Riccati equation is converted into a continuous one, which can be solved by using the reduced-order recursive method already documented in the control literature. This method produces the reduced-order near-optimal solution up to an arbitrary order of accuracy and reduces the size of required computations. The method is very suitable for parallel programming. A real world example, an F-8 aircraft, demonstrates the efficiency of the proposed method.

Multiple models, multiplicative noise and linear quadratic control—algorithmic aspects

Numerical solution of certain linear quadratic (LQ) control problems for robust design is addressed. An iterative method is suggested and analysed for solving a minimax multiple model LQ control problem. A convergent iterative method is studied for finding the constant feedback gains of a given linear controller so as to solve a spectral radius functional minimization problem for multiple plants. Furthermore, a convergent iterative method is proposed for solving the maximum entropy parametric LQ control problem with multiplicative noise.

Convergence Rates for the Feedback Operators Arising in the Linear Quadratic Regulator Problem Governed by Parabolic Equations

The linear quadratic control problem for systems governed by evolution equations of parabolic type is considered. The optimal feedback control uses a Riccati operator which is approximated by a Galerkin scheme. Optimal convergence rates (except for a logarithmic factor) are derived for the approximation of the Riccati operator for both the finite and infinite time interval. This is done in an abstract Hilbert space framework assuming a certain rate of convergence for the uncontrolled system. In the numerical part, the Galerkin method is compared with the collocation method with special emphasis on the feedback gains.

Parallel reduced-order controllers for stochastic linear singularly perturbed discrete systems

An approach to the decomposition and approximation of linear quadratic Gaussian control problems for singularly perturbed discrete systems at steady state is presented. The global Kalman filter is decomposed into separate reduced-order local filters through the use of a decoupling transformation. A near-optimal control law is derived by approximating coefficients of the optimal control law. The proposed method allows parallel processing of information and reduces offline and online computational requirements. A real-world example demonstrates the efficiency of the proposed method. >

A Unified Approach to Optimal Feedback in the Infinite-Dimensional Linear–Quadratic Control Problem with an Inequality Constraint on the Trajectory or Terminal State

This paper develops a unified approach to optimal feedback control in the infinite-dimensional linear-quadratic control problem with an inequality constraint on the trajectory or terminal state. A general abstract Hilbert-space framework for the problem is provided, and then the results are specified to concrete optimal problems for a linear infinite-dimensional system which allows unboundedness in the input and output operators. In all cases, the same technique is used in order to obtain the optimal control in feedback form; in some cases, the differential Riccati equation is derived for the optimal cost operator.

On the computation of the main eigen-pairs of the continuous-time linear quadratic control problem

The degeneration of the eigenvalue equation of the discrete-time linear quadratic control problem to the continuous-time one when Δt→0 is given first. When the continuous-time n-dimensional eigenvalue equation, which has all the eigenvalues located in the left half plane, has beeh reduced from the original 2n-dimensional one, the present paper proposes that several of the eigenvalues nearest to the imaginary axis be obtained by the matrix transformation Ae=eA. All the eigenvalues of Ae are in the unit circle, with the eigenvectors unchanged and the original eigenvalues can be obtained by a logarithm operation. And several of the eigenvalues of Ae nearest to the unit circle can be calculated by the dual subspace iteration method.

Near-optimum steady state regulators for stochastic linear weakly coupled systems

This paper presents an approach to the decomposition and approximation of the linear quadratic Gaussian estimation and control problems for weakly coupled systems. The global Kalman filter is decomposed into separate reduced-order local filters via the use of a decoupling transformation. A near-optimal control law is derived by approximating the coefficients of the truly optimal control law. The order of approximation of the optimal performance is O( ε N ), where N is the order of approximation of the coefficients. A real world power system example demonstrates the failure of O( ε 2) and O( ε 4) approximations and the necessity for the existence of the O( ε N ) theory. The proposed method produces the reduction in both off-line and on-line computational requirements and leads to convergence under mild assumptions. In addition, only low-order systems are involved in algebraic calculations and no analyticity requirement (a standard assumption for the power series method) is imposed on system coefficients.

An Approximation Theorem for the Algebraic Riccati Equation

For an infinite-dimensional linear quadratic control problem in Hilbert space, approximation of the solution of the algebraic Riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems. As an application, strong onvergence of the approximating Riccati operators in case of a previously developed spline approximation scheme for delay systems is established. Finally, convergence of the transfer-functions of the approximating systems is investigated.

Factorization and the Synthesis of Optimal Feedback Gains for Distributed Parameter Systems

An approach based on Volterra factorization leads to a new methodology for the analysis and synthesis of the optimal feedback gain in the finite-time linear quadratic control problem for distributed parameter systems. The approach circumvents the need for solving and analyzing Riccati equations and provides a more transparent connection between the system dynamics and the optimal gain. The general results are further extended and specialized for the case where the underlying state is characterized by autonomous differential-delay dynamics. Numerical examples are given to illustrate the second-order convergence rate that is derived for an approximation scheme for the optimal feedback gain in the differential-delay problem.

Infinite networks and quadratic optimal control

A discrete-time quadratic optimal control problem is shown to be equivalent to an infinite cascade ofn-port electrical networks. Both problems can be expressed as a generalized Schur complement (or shorted operator) of a block tridiagonal operator. Using Maxwell's principle for resistive networks, a variational formulation of the cascade limit ofn-port networks is given. For one special case a formula for the solution of the quadratic control problem is given in terms of the geometric mean of operators.

Computational structural mechanics. Optimal control and semi-analytical method for PDE

Based on the generalized variational principle the analysis of a substructural chain is considered, and the 1:1 relationship between the structural analysis problem and the linear quadratic optimal control problem is then introduced. Hence, the algebraic Riccati equation can be solved in two ways; the upper-bound and lower-bound iterative methods. The theory and methods of structural analysis problems can then be transferred to the linear quadratic optimal control problems. As to the continuous coordinate, and/or continuous-time problems, it can be shown that the linear quadratic control problem also corresponds to the semi-analytical method of the elliptic partial differential equation. It is hoped that the unified method of these disciplines will lead to further progress.

Quadratic Control for Stochastic Systems Defined by Evolution Operators and Square Integrable Martingales

The infinite dimensional version of the linear quadratic cost control problem is studied by Curtain and Pritchard [2], Gibson [5] by using Riccati integral equations, instead of differential equations. In the present paper the corresponding stochastic case over a finite horizon is considered. The stochastic perturbations are given by Hilbert valued square integrable martingales and it is shown that the deterministic optimal feedback control is also optimal in the stochastic case. Sufficient conditions are given for the convergence of approximate solutions of optimal control problems.